# Distance Word Problems

In these lessons, we will learn how to solve rate time distance word problems where the objects are traveling in opposite directions. You may be required to find the time when the objects meet or the time when the objects are a certain distance apart.

Distance problems are word problems that involve the distance an object will travel at a certain average rate for a given period of time.

The formula for distance problems is: distance = rate × time or d = r × t.

Things to watch out for:

Make sure that you change the units when necessary. For example, if the rate is given in miles per hour and the time is given in minutes then change the units appropriately.

It would be helpful to use a table to organize the information for distance problems. A table helps you to think about one number at a time instead being confused by the question.

The following diagrams give the steps to solve Rate Time Distance Word Problems. Scroll down the page for examples and solutions. ### Distance Problems: Traveling In Opposite Directions

Example:
A bus and a car leave the same place and traveled in opposite directions. If the bus is traveling at 50 mph and the car is traveling at 55 mph, in how many hours will they be 210 miles apart?

Solution:
Step 1: Set up a rtd table.

r bus car

Step 2: Fill in the table with information given in the question.

If the bus is traveling at 50 mph and the car is traveling at 55 mph, in how many hours will they be 210 miles apart?

Let t = time when they are 210 miles apart.

r t d bus 50 t car 55 t

Step 3: Fill in the values for d using the formula d = rt

 r t d bus 50 t 50t car 55 t 55t

Step 4: Since the total distance is 210, we get the equation:

50t + 55t = 210
105t = 210
Isolate variable t Answer: They will be 210 miles apart in 2 hours.

Example of a distance word problem with vehicles moving in opposite directions

In this video, you will learn to solve introductory distance or motion word problems - for example, cars traveling in opposite directions, bikers traveling toward each other, or one plane overtaking another. You should first draw a diagram to represent the relationship between the distances involved in the problem, then set up a chart based on the formula rate times time = distance.

The chart is then used to set up the equation.

Example:
Two cars leave from the same place at the same time and travel in opposite directions. One car travels at 55 mph and the other at 75 mph. After how many hours will they be 520 miles apart?

Rate-Time-Distance Problem

Solve this word problem using uniform motion rt = d formula:

Example:
Two cyclists start at the same corner and ride in opposite directions. One cyclist rides twice as fast as the other. In 3 hours, they are 81 miles apart. Find the rate of each cyclist.

Distance - Opposite Directions

Example:
Brian and Jennifer both leave the convention at the same time traveling in opposite directions. Brian drove at 35 mph and Jennifer drove at 50 mph. After how much time were they 340 miles apart?

Distance - Opposite Directions find t

Example:
Two joggers start from opposite ends of an 8 mile course running towards each other. One jogger is running at a rate of 4 mph. The other is running at a rate of 6 mph. After how long will the joggers meet?

Distance - Opposite Directions find r

Example:
Bob and Fred start from the same point and walk in opposite directions. Bob walks 2 mph faster than Fred. After 3 hours they are 30 miles apart. How fast does each walk?

GMAT Challenge Question: Distance/Rate/Time

Example:
Trains A and B left stations R and S simultaneously on two separate parallel rail tracks that are 350 miles long. The trains pass each other at point X after traveling for a certain amount of time. How many miles of the rail tracks has train A traveled when the two trains passed each other?

1. Up to point X, the average speed of train B was 25% less than the average speed of train A.
2. Up to point X, the average speed of train B was 60 mph and it took two and a half hours for train B to arrive at point X.

Try the free Mathway calculator and problem solver below to practice various math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations. 